The velocity field of turbulent flows is three-dimensional, time-dependent and chaotic (Pope, 2000). The difficulty of computing and modeling turbulent flows lies in the broad range of spatial and temporal scales. The largest turbulent motions of size L depend upon geometry and boundary conditions. The smallest dissipative motions are characterized by the Kolmogorov length scalelη. For isotropic turbu-lence, the ratio of both length scalesL/lη is proportional to (Reynolds number)3/4, cf. Pope (2000), which provides a direct link to the grid resolution that is required to resolve the Kolmogorov dissipation scale. The Direct Numerical Simulations (DNS) of the governing equations, Eqs. (2.1)-(2.5), requires the resolution of all length and time scales (because no turbulence model is applied). Rough estimates suggest that turbulent flows at high Reynolds number are therefore out of reach for DNS – even with todays computational capacities.
For flows at high Reynolds numbers Large-Eddy Simulation (LES) is becoming a more widely used simulation tool – also outside the realm of academic research.
In LES, the grid on which the governing equations are solved numerically is too coarse to represent the entire range of turbulent length scales.9 As a consequence, the energy transfer between the smallest and largest scales is not resolved and a subgrid-scale (SGS) model is required to represent the influence of non-resolved turbulent scales. Following the argument of Hickel et al. (2014),e.g., only the large resolved scales provide a direct representation of the energy-containing flow structures, which in fact is adequate for most practical purposes in engineering.
These energy-containing large motions are directly affected by boundary conditions and hence are not universal.
The numerical solution of the conservation equations implicitly generates a scale separation through both the use of a computational grid of finite spacing and through the numerical discretization of continuous operators. Following the ap-proach of Leonard (1975), this implicit scale separation can also be expressed as a convolution of the governing equations with a homogeneous filter kernel yielding
9The smallest temporal scales are typically assumed to be resolved by choosing a sufficiently small time step (Adams et al., 2004)
2.4 Numerical Method
the filtered Navier-Stokes equations, see e.g., Garnier et al. (2009) for a more de-tailed discussion. Filtering of nonlinear terms in the Navier-Stokes equations leads to unclosed SGS terms, which need to be modeled. There are explicit and implicit SGS models. SGS effects are modeled explicitly if the filtered governing equations are augmented by additional terms to account for SGS effects and the resulting system of equations is subsequently discretized. Standard explicit SGS models rely on the assumption that turbulence model and numerical dissipation from trunca-tion errors do not interfere. At grid resolutrunca-tions that are typically used for LES, this basic assumption is not necessarily met and the truncation error of the dis-cretization scheme can dominate SGS-model terms, see, e.g., Adams et al. (2004) and Hickel et al. (2014), and references therein. Implicit LES (ILES) approaches, on the other hand, make use of the fact that numerical truncation errors resulting from the discretization of the unmodified governing equations can in principle act as SGS model. Here, the difficulty is to modify the truncation error in such a way that dissipation and diffusion of resolved flow scales is consistent with flow physics.
In this work, the Adaptive Local Deconvolution Method (ALDM) is used. ALDM is a nonlinear finite-volume (FV) scheme for ILES and has been developed for in-compressible (Adams et al., 2004; Hickel et al., 2006) and in-compressible turbulence (Hickel et al., 2014). The method operates on the discretization of the hyperbolic flux which is mainly responsible for SGS effects. Free parameters that result from the the reconstruction procedure (approximation of the unfiltered solution at cell faces) and secondary regularization within the numerical flux function are used to calibrate the spatial truncation error of the FV scheme such that a physically mo-tivated turbulence model consistent with turbulence theory is obtained. A much more detailed description of the original ALDM framework can be found in Adams et al. (2004), Hickel et al. (2006) and Hickel et al. (2014).
Due to the large density gradients that are present in real-gas flows as considered in this work, a modification of the original ALDM scheme as documented in Hickel et al. (2014) was necessary to enhance numerical stability. At contact discontinu-ities, numerical oscillations were observed for the mass density ρ, partial densities ρYi and internal energy e. When converting conservative to primitive variables in the non-linear cubic EOS framework, severe pressure and temperature oscillations occurred causing a blow-up of the numerical solution. The following modifications led to a significant improvement of the numerical stability: 1. A second-order upwind biased numerical flux function together with the van Albada limiter (van Albada et al., 1982) for the advective transport of mass and internal energy re-moved all oscillations and numerically computed values of the species mass fraction Yi from (ρYi)/ρ were bounded in the interval [0,1]. 2. Any contribution of the shock sensor to the dissipation matrix, cf. Eq. (37) in Hickel et al. (2014), was
omitted since the upwinding itself introduced a sufficient amount of dissipation.
The viscous flux of the governing equations is discretized with a2nd order central difference scheme, and the3rd order explicit Runge-Kutta scheme of Gottlieb and Shu (1998) is used for time integration. The left-hand side of the pressure-evolution equation of the quasi-conservative method, Eq. (2.5), is discretized consistently with the transport of internal energy such that both discretizations are identical up to machine precision for a single-species perfect gas (Nc = 1, cv 6= f(T) and p= (cp/cv−1)ρe).
Chapter 3
LN2-GH2 Shear Coaxial Flow
In this chapter, we will study a selected operating point of a series of experiments of Oschwald et al. (1999), in which quantitative density measurements in coaxial liquid nitrogen (LN2) and gaseous hydrogen (GH2) jets at supercritical pressures (with respect to the critical pressure of pure nitrogen and pure hydrogen) were obtained. Emphasis is placed on both a quantitative and qualitative compari-son between experimental and numerical data and the assessment of uncertainties related to both of them. In particular, two-phase phenomena that occur at the present operating conditions and the accuracy of the thermodynamic modeling approach will be addressed. All results presented in this chapter are simulated with the flow solver INCA and the numerical framework introduced in Chapter 2.
The interested reader may find additional results for this test case obtained with other CFD codes and volume translated EOS in Müller et al. (2015), Matheis et al.
(2015) and Müller (2016).
Major parts of this chapter are based on the author’s article Large-eddy simulation of cryogenic jet injection at supercritical pressures.
In J. Bellan (Eds.) High Pressure Flows for Propulsion Applica-tions, Progress in Astronautics and Aeronautics (under review).
3.1 Problem Description
We study the coaxial injection of hydrogen and nitrogen into a supercritical nitro-gen atmosphere, which resembles conditions encountered in real rocket engines. By substituting oxygen with the inert gas nitrogen (with its fluid mechanical proper-ties being not too different from oxygen), this setup allows for an isolated view on binary mixing processes near the injector without the influence of chemical reac-tions (Oschwald et al., 1999). The setup has been investigated experimentally by Oschwald et al. (1999) who performed a series of 2D-Raman density measurements with particular emphasis on atomization and mixing mechanisms. The chamber pressure is4MPa, thus, supercritical with respect to the critical pressures of pure nitrogen (pc,N2 = 3.34 MPa) and pure hydrogen (pc,H2 = 1.29 MPa). The mea-surement campaign covered experiments with and without the coaxial injection of hydrogen at several injection temperatures and velocities. To our knowledge, this is currently the only published experimental study on inert coaxial injection at LRE relevant conditions that obtained quantitative data with laser diagnostic methods. Data that have been obtained in a similar measurement campaign by Decker et al. (1998) have unfortunately never been published.
In the following, we will discuss LES results achieved for the operating conditions E4, which are summarized in Tab 3.1. Hydrogen (outer stream) and nitrogen (inner stream) are injected through a coaxial injector element into a cylindrical tank (D = 10 cm) filled with nitrogen at 4 MPa and 298.15 K. The inner and outer diameter of the hydrogen annulus areDH2,i = 2.4 mm andDH2,o = 3.4mm, respectively; the inner nitrogen injector isDi = 1.9mm in diameter. A schematic of the experimental and optical setup is given in Fig. 3.1. The (ρ, T) diagram shown in Fig. 3.2 illustrates the nominal operating conditions for the main nitrogen injection. With a temperature ofTN2 = 118K, nitrogen is initially in a compressed liquid state for operating point E4 (ρN2 = 584.43 kg/m3). As seen from Fig. 3.2, the PR EOS, which will be used in the subsequent simulations yields a reasonably good approximation of the (pure) nitrogen inflow density with an error of about
∼4%compared to the NIST reference data. The nominal bulk velocity of nitrogen is uN2 = 5 m/s and hydrogen is injected with a bulk velocity of uH2 = 120 m/s.
3.1 Problem Description
Fig. 1.Sketch of the cryo-injector test chamber allowing concentration measurements by spontaneous Raman
scattering.
In former experiments the disintegration of the LN2jet has been investigated by shadowgraphy by our group (Mayer et al.
1996). These experiments provided information on the quali-tative change of the disintegration phenomenology from a spray like atomization at subcritical pressures to a behaviour more similar to a free gas jet at supercritical pressures. At subcritical pressure the shadowgraphs show a well defined jet surface with large scale disturbances while going to supercriti-cal pressure the images show a more diffuse jet surface and decreasing length scales of the disturbances.
The injected cryogenic nitrogen jet has a much higher density than the nitrogen gas in the flow channel at ambient temperature. The jet disintegration process can therefore be analyzed by measuring the spatial density distribution down-stream the injector. Since powerful pulsed UV-lasers are available, quantitative species detection with spontaneous Raman scattering has become a standard diagnostic technique to analyse mixing and combustion in gas flows at atmospheric pressure (Drake et al. 1981; Dibble et al. 1984; Pitz et al. 1990;
Brockhinke et al. 1995; Wehrmeyer et al. 1995). Due to the 1/!4 dependence of the Raman cross section on the wavelength
!even temporally resolved 1D-measurements are possible in the UV. In high pressure applications the Raman photon statistics benefit from the high number density of the molecu-les. The potential of Raman scattering for the investigation of high pressure flows has been demonstrated by several groups.
Anderson et al. (1995) applied 2D-Raman imaging for density measurements of supercritical LOX-droplets in He. They report the problem of gas break down in the inhomogeneous two phase flow. Woodward and Talley (1996) investigated the structural differences between injecting a transcritical cryogenic nitrogen jet into helium or nitrogen by 2D Raman imaging. Quantitative 1D-Raman multispecies measurements in a GOX/GH2-flame has been demostrated by Farhangi et al.
(1994) and Foust et al. (1996). Using a flashlamp-pumped dye laser Foust et al. acquired temporally averaged 1D-density profiles at a pressure of 6.9 MPa. Wehrmeyer et al. (1997) did 1D-multispecies measurements in a LOX/GH2flame at pres-sures up to 6.04 MPa. For the measurements the beam of a narrow band KrF-excimer laser had to be used unfocussed to avoid the generation of stimulated Raman signal. Yeralan et al.
(1997) used a doubled Nd : YAG-laser for 1D-density measure-ments in a LOX/GH2-combustor. In these experimeasure-ments inves-tigating LOX/GH2 two-phase flows the measurement volume was about 30 LOX-post diameters downstream of the injector.
At this location density gradients and hence refraction index gradients have already been decreased by mixing and heat transfer.
In high pressure flows of interest there are always species and/or temperature gradients resulting in spatial and temporal variations of the index of refraction. This is especially true in the near injector region which is of interest in the experiments presented here. The density of the injected cryogenic nitrogen can reach 700 kg/m3, about 16 times higher than that of the reservoir gas at ambient temperature. At the refractive index gradients the high power flux of pulsed lasers in the measure-ment volume (typically in the order of GW/cm2) may be focused and increased substantially resulting in gas break
down or the stimulation of non linear interaction of the laser wave with the molecules. The proof of Raman signal linearity in homogeneous gas mixtures will not guarantee that stimu-lated Raman scattering will not occur in a turbulent flow with high density gradients. Temporally resolved 2D-density measurements with a XeF-laser have been done by Decker et al.
(1998) at the cryo-injector test facility. Care has been taken in these experiments to avoid non-linear signal generation.
Nevertheless background signal has been detected interfering with the Raman signal. To rule out any possibility of gas break down or the generation of stimulated Raman processes, we use a cw-laser in the measurements presented here resulting in a power flux in the measurement volume of about 10 KW/cm2.
A further motivation for the experiment was that the experi-mental set up using a cw-laser is not as complex as using an excimer-laser. This is resulting in short set up times at the test facility and thus in flexibility in test facility operation. The price to pay is the loss of temporal resolution since the reduced Raman signal intensity requires data acquisition times in the order of 1s.
2
Cryo-injector test facility
A sketch of the test chamber is shown in Fig. 1. Details of the test facility are given in Mayer et al. (1996). A cylindrical reservoir with 10 cm diameter can be pressurized with nitrogen up toP!!6 MPa, a factor of 1.76 above the critical pressure of nitrogenPc,N2!3.4 MPa. The nozzle is mounted to inject the fluids parallel to the flow channel. The injector can be translated in the direction of the flow allowing optical diagnostics of the jet at arbitrary positions downstream the injector through four windows.
The dimensions of the coaxial injector are chosen to be representative for injection conditions in rocket engines.
Nitrogen is injected through the central post ofd!1.9 mm diameter and 22 mm length. Injection of hydrogen through an annular slit of 2.4 mm and 3.2 mm inner and outer diameter respectively is optional. Experiments presented here were done with a nitrogen free jet without coaxial injection of H2. The velocity of the injected fluid is calculated on the basis of the 498
(a)Schematic of the experimental setup. Fig. 3.Optical set-up
Table 1.Injection conditions and
similarity numbers Test P! v" T" P!/P# !"/!! Re"!10!3 j"!"v2"!10!3
cases [MPa] [m/s] [K] [kg/m/s2]
A4 4 5 140 1.17 3.34 115 3.7
Fig. 2.Density and specific heat of nitrogen as function of temperature. Injector exit conditions for the test cases (see Table 1) are marked
measured mass flux. The temperature of the reservoir is controlled to be stationary at 298 K by electrical heating. The temperature of nitrogen at the injector exit is measured by a thermocouple mounted 20 mm upstream from the injector exit. The temperature of the feeding line is not actively controlled and is the result of the heat flux balance in the flow system.
3 Test matrix
In the experiments presented here the cryogenic jet disintegra-tion has been analyzed at several injecdisintegra-tion condidisintegra-tions defined by injection temperature and injection velocity.
The dependence of the nitrogen density!on temperature Tis shown in Fig. 2. The data for this figure are taken from Younglove (1982). As can be seen in Fig. 2 near 130 K at 4 MPa and near 140 K at 6 MPa!is a sensitive function ofT. The specific heatcpwhich is divergent at the critical point shows at supercritical pressures a maximum atT*where"!/"Tis maximal. It is obvious that nitrogen injected at the conditions of our test cases can be treated adequately neither as a gas nor as a liquid.
The test cases that have been investigated are summarized in Table 1. To analyze the influence of the density of the reservoir gas on the atomization process the investigations have been
done at two pressure levels in the flow channel at 4 MPa and at 6 MPa, 1.17 and 1.76 times the critical pressure of nitrogen.
In test cases A and B nitrogen was injected with the same injection velocity (5 m/s) but at different injection temper-aturesT"of 140 K and 118 K respectively. At 4 MPac$is maximal atT*"129.5 K. It isT"#T*for case A andT"$T*for test case B. Decreasing the injection temperature the ratio of the gas density in the flow channel!!and the density of the cryogenic jet!"at the injector exit!"/!!increases strongly from 3.34 (case A, 140 K) to 12.5 (case B, 118 K). The exit temperature in case B is the equilibrium temperature resulting from the heat flux balance in the supply lines at the LOX mass flow of the test case. The measurements for case A have been done during the transient cooling down of the feeding lines from ambient temperature to the equilibrium temperature of 118 K. Measurements have been done when the thermocouples measured 140 K in the LN2post.
In test case C cryogenic nitrogen is injected at similar density as in case B but at a higher velocity (20 m/s). The exit temperature of 100 K in case C is the equilibrium temperature for the nitrogen massflow corresponding to 20 m/s injection velocity. In case C as in case B isT"$T*.
4
Optical set-up
A sketch of the optical set-up shown in Fig. 3. The 488 nm line of an Ar%-laser has been used to excite the Raman transitions.
The laser was linearly polarized with theEo-vector perpendicu-lar to the scattering plane. The laser power was 1.5 W.
499
(b) Schematic of the optical setup.
Figure 3.1: Schematic of the experimental and optical setup used in the cam-paign of Oschwald and Schik (1999). Reprinted with permission from Springer, Fig. 1 and Fig. 3, Copyright 2017, Springer-Verlag Berlin Heidelberg.
These velocities have been calculated from LN2 and GH2 mass flow rates, which were measured directly with a Coriolis mass flowmeter (private communication with Michael Oschwald).
Table 3.1: Test case definition. a Nominal experimental operating conditions ac-cording to Oschwald et al. (1999). b Calculated using NIST with nom-inal temperature and pressure. c Calculated using the PR EOS with nominal temperature and pressure. d Directly measured with a Cori-olis mass flowmeter (private communication with Michael Oschwald).
e Measured in the experiment on the jet centerline, cf. Fig. 3.5a and Fig. 12 in Oschwald et al. (1999). f Calculated from m˙ N2 and ρN2. g Calculated from ρN2 and pusing the PR EOS.
p[MPa] TN2 [K] uN2 [m/s] ρN2 [kg/m3] m˙ N2 [g/s]
105 110 115 120 125 130 135 140 145 0
200 400 600
ρExpE4 = 390.18 kg/m3 ρ(TE4Exp) = 584.43 kg/m3
TE4Exp= 118 K
T(ρExpE4 ) = 128.8 K
∆ρ∼194 kg/m3
∆T ∼11 K
T [K]
ρ[kg/m3]
Figure 3.2: Thermodynamic conditions of the main nitrogen injection for test case E4-T118 and E4-T128 at 4MPa. ( ) PR EOS;◦Reference data is taken from the NIST (Lemmon et al., 2013)